I wouldnt give that info myself either, but I don't think there's anything skeezy going on. I'll bet the recruiter wants that info to see if you pass a background check before handing you over to the hiring company.
They want to run a credit and driving records check.
I wouldn't give this info.
I've never once been asked for this info by a recruiter. Any background or credit checks that been conducted on me in the course of finding employment have always been done by the hiring company, never the recruiter.
Personally, I would consider requiring this kind of info a Red Flag.